Mathematics Problem Archive

(Swarup) Suppose $H$ is a finitely presented subgroup of a word-hyperbolic group $G$ and has finite index in its normalizer. Assume that there is $n>0...

(Whyte) Let Γ be a 1-ended hyperbolic group which is not virtually a surface group. Can every infinite index subgroup be free?...

Is every word-hyperbolic group residually finite?...

(Dani Wise) Let $G^n$ denote the Cartesian product of $n$ copies of $G$, and let $\operatorname{rank}(G^n)$ be its smallest number of generators. If $...

(Dani Wise) Find 'nice' groups, for example CAT(0) or automatic groups, for which $\operatorname{rank}(G^n)$ does not tend to infinity as $n\to\infty$...

(Ian Leary) Is there a version of the Kan-Thurston theorem using only CAT(-1) groups, or word hyperbolic groups? (The statement should be: for any fin...

(Gromov) If $G$ admits a finite-dimensional $K(G,1)$, does $G$ act properly discontinuously by isometries on a complete CAT(0) space?...

Suppose a group G acts properly discontinuously and cocompactly by isometries on two CAT(0) spaces X and Y . Croke-Kleiner have examples where the bou...

Do CAT(0) (or (bi)automatic) groups satisfy the Tits alternative?...

(Eric Swenson) Let $X$ be a proper CAT(0) metric space and $G$ a finitely generated group acting properly discontinuously by isometries on $X$. (1) Ca...

(Ruth Charney) Classify Coxeter groups up to isomorphism....

(Ruth Charney) Classify Artin groups up to isomorphism....

(Ruth Charney) Are all [finite type] Artin groups CAT(0)?...

(Ruth Charney) Are all Artin groups automatic?...

(J. Cornick) If G is f.g. and the homological dimension hd G = 1, is G free?...

Characterize 1-relator groups which are limit groups....

(Swarup) Suppose G is a 1-ended finitely presented group that acts on a compact connected metric space X as a convergence group. What can be said abou...

(Jim Anderson) If G is a group of isometries of Hn, denote by Ax(G) the set of axes of the elements of G. If G1 and G2 are finitely generated and disc...

(Misha Kapovich) Is there a finitely-generated discrete subgroup of SO(n, 1) whose action on the limit set is not ergodic? Is not recurrent?...

(Ian Leary) Is there a group of finite vcd that does not act with finite stabilizers on an acyclic complex of dimension equal to its vcd?...

(Peter Kropholler) If G is FP over the rationals, is there a bound on the orders of finite subgroups of G?...

(Sageev) Is there a f.p. 1-ended group G with G ∼= G∗Z?...

(Papasoglou) Is there a f.p. torsion-free group G that does not split over a virtually abelian subgroup, but has infinitely many splittings over F2?...

Can Zp∞be embedded in an FP∞-group? Or in an FP3-group? Can an Fn-group be embedded in an Fn+1-group (n ≥2)?...

Is there a finitely presented group $G=F/N$, with $F$ a finite-rank free group, such that $$d_G\!\left(N/[N,N]\right)<d_F(N)<\infty?$$ Here $d_F(N)$ i...

(Wise) Is there a finite aspherical 2-complex X with π1(X) coherent and with χ(X) ≥2?...

(Igor Belegredek) Let X be a non-positively curved symmetric space. Find conditions on a group Γ so that the space of conjugacy classes of faithful di...

(Kleiner) What are the q.i’s of the Gromov-Thurston examples of negatively pinched manifolds?...

(Feighn) Let $\phi:F_n\to F_n$ be an automorphism and let $M_\phi$ be its mapping torus. Classify these groups up to quasi-isometry....

Does Out(Fn) (n > 2) have a right orderable subgroup of finite index?...

Do there exist f.g. free subgroups of MCG(S) consisting of identity and pseudo-Anosov mapping classes which are not Schottky?...

For $\operatorname{Out}(F_n)$, do there exist finitely generated free subgroups consisting of the identity and irreducible automorphisms which are not...

For $\operatorname{Out}(F_n)$, is the natural map $\operatorname{Out}(F_n)\to QI(\operatorname{Out}(F_n))$ an isomorphism? Does every quasi-isometry o...

Is MCG(Sg,b,n) linear?...

(Kevin Whyte) Let K be a finite complex with π = π(K) amenable. Is there a uniform bound to the betti numbers of finite covers of K?...

(S. Ivanov) Is there a f.p. slender group which is not polycyclic-by-finite?...

Let $G$ be a one-relator group whose relator $W$ is a cyclically reduced word in the generators, and let $P$ be the submonoid of $G$ generated by all ...

Let a finitely presented pro-$p$ group have $r+2$ generators and $r$ relators. Must it have an open subgroup that maps continuously onto a nonabelian ...

Can the sequence of metric balls in an infinite Cayley graph form a family of expanders?...

Let $\Gamma$ be finitely generated and let $\{H_n:n\geq1\}$ be a property-$\tau$ family of finite-index normal subgroups. Does the chain $\Gamma_n=\bi...

Let $\Gamma$ be finitely presented with a chain of finite-index normal subgroups having trivial intersection and property $\tau$. Must $\Gamma$ contai...

For a graph sequence $(G_n)$ define $e((G_n))=\liminf |E(G_n)|/|V(G_n)|$, and define its combinatorial cost as the infimum of $e((H_n))$ over graph se...

Compactness implies that for every $\varepsilon>0$ there is $K>0$ such that every finite graph can be approximated within error $\varepsilon$ by a fin...

Which probability measures can occur as eigenvalue distributions of finite $d$-regular graphs? Find natural restrictions....

Do uniformly random $d$-regular graphs converge, in local-global convergence, to the weak closure of independent identically distributed processes?...

Is the i.i.d. action of the free group $F_2$ a local-global limit of finite actions of $F_2$?...

Can every ergodic unimodular random network that is almost surely an infinite tree be obtained as the limit of an expander family?...

Let $G$ be an infinite vertex-transitive graph, let $A$ be a finite vertex set, let $b$ be a vertex, and let $\partial A$ be the set of vertices at di...

Define the first $L^2$ Betti number of a vertex-transitive graph $G$ from the expected degree of a free spanning forest. Do $G$ and its square $G^2$ h...

Does every infinite Cayley graph, or every infinite vertex-transitive graph, have a spanning tree without leaves?...